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We evaluated <P(e, 0)> for 12 cases of e between 0 and 6: e=0.0, 0.01, 0.1, 0.5, 0.75, 0.9, 1.0, 1.2, 1.5, 2.0, 4.0, and 6.0. As for r_p, we considered three cases: r_p=0.005, 0.001, and 0.0002. These are representative values of radii of protoplanets at the Earth, Jupiter, and Neptune orbits regions, respectively. The numbers of collision orbits found by our orbital calculation are shown in Table 3 for representative values of e. From Table 3 we can expect the statistical errors in the evaluated collisional rate to be within 5% for the cases of e≦1.5 and within 8% for e=4 and 6; they are smaller than that of the previous studies by Nishida (1983) and by Wetherill and Cox (1985). The calculated collisional rate is summarized in terms of the enhancement factor defined by Eq. (27) and shown in Fig.11, as a function of e and r_p. From Fig.11 one can see that the collisional rate is always enhanced by the effect of solar gravity, compared with that of the two-body approximation <P(e,0)>_2B. In particular, in regions where e≦1, R(e,0) is almost independent of e, having a value as large as 3. At e≦1, R(e,0) has a notable peak beyond which the enhancement factor decreases gradually with increasing e. For large values of e, i.e., e≧4, <P(e,0)> tends rapidly to <P(e,0)>_2B. As seen in the next section, we will find a similar dependence on e even in the three-dimensional case (i≠0) as long as we are concerned with cases where i≦2. お手数ですが、よろしくお願いします。
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0と6の間の12通りのeについて<P(e, 0)>を評価した。それぞれのeは0.0, 0.01, 0.1, 0.5, 0.75, 0.9, 1.0, 1.2, 1.5, 2.0, 4.0, そして 6.0 である。r_pに関しては3つのケースを考慮した。すなわち r_pが0.005, 0.001, そして 0.0002の場合である。これらはそれぞれ地球、木星、海王星の軌道領域における原始惑星の半径の代表的数値である。我々の軌道計算により発見された衝突軌道の数はeの代表数値に基づいて表3に示してある。表3により、評価した衝突速度の統計的過誤は、eが1.5と同じかそれよりも小さい場合は5％以内,eが4 および6と同じ場合は8％以内と想定できる。それらは西田（1983年）やウェザリルとコックス（1985年）による過去の研究の数値よりも小さい。 算出された衝突速度は方程式(27）で定義された促進係数を使って要約し、 eとr_pの関数として図11に示した。図11から衝突速度は二体近似<P(e,0)>_2Bのものと比較して、常に太陽重力の影響により強められていることがわかる。特に eが1と同じかそれよりも小さい領域では R(e,0)は常にeとはほとんど無関係で3程度の数値をもつ。eが1と同じかそれよりも小さいところで、R(e,0)は促進係数がeの増加と共に徐々に減っていく前の顕著なピークを有する。eが4と同じかそれよりも大きいといった大きな数値では、 <P(e,0)>は急速に<P(e,0)>_2Bへと近づいてゆく。次章でわかるようにiが2と同じかそれよりも小さい場合を念頭におく限り、3次元(iが0ではない)においてもeへの類似の依存性を発見するのだ。
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In Fig.13, we compare our results with those of Nishiida (1983) and Wetherill and Cox (1985). Nishida studied the collision probability in the two-dimensional problem for the two cases: e=0 and 4. For the case of e=0, his result (renormalized so as to coincide with our present definition) agrees accurately with ours. But for e=4, his collisional rate is about 1.5 times as large as ours; it seems that the discrepancy comes from the fact that he did not try to compute a sufficient number of orbits for e=4, thus introducing a relatively large statistical error. The results of Wetherill and Cox are summarized in terms of v/v_e where v is the relative velocity at infinity and v_e the escape velocity from the protoplanet, while our results are in terms of e and i. Therefore we cannot compare our results exactly with theirs. If we adopt Eq. (2) as the relative velocity, we have (of course, i=0 in this case) (e^2+i^2)^(1/2)≒34(ρ/3gcm^-3)^(1/6)(a_0*/1AU)^(1/2)(v/v_e). (34) According to Eq. (34), their results are rediscribed in Fig.13. From this figure it follows that their results almost coincide with ours within a statistical uncertainty of their evaluation. 7. The collisional rate for the three-dimensional case Now, we take up a general case where i≠0. In this case, we selected 67 sets of (e,i), covering regions of 0.01≦i≦4 and 0≦e≦4 in the e-i diagram, and calculated a number of orbits with various b, τ,and ω for each set of (e,i). We evaluated R(e,i) for r_p=0.001 and 0.005 (for r_p=0.0002 we have not obtained a sufficient number of collision orbits), and found again its weak dependence on r_p (except for singular points, e.g., (e,i)=(0,3.0)) for such values of r_p. Hence almost all results of calculations will be presented for r_p=0.005 (i.e., at the Earth orbit) here. Fig.13. Comparison of the two-dimensional enhancement factor R(e,0) with those of Nishida (1983) and those of Wetherill and Cox (1985).Their results are renormalized so as to coincide with our definition of R(e,0). 長文ですが、よろしくお願いします。
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Figure 8 shows r_min in the second encounter for b=2.8. In this case, there are four zones of close-encounter orbit in the τ-ω diagram. Comparing Fig.8 with Fig. 7b, the total area occupied by the recurrent close-encounter orbits (the dotted regions in Fig. 8) is smaller than that in the first encounter but not small enough to be neglected. Collision orbits belong necessarily to close-encounter orbits. Consequently, to find collision orbits, we subdivided the τ-ω phase space of close-encounter orbits (i.e., the finely dotted regions in Fig.7) more densely (mesh width being as small as 0.002π in τ) and pursued orbits for each set of τ and ω. Furthermore, as the phase volume of τ and ω occupied by collision orbits, we evaluated a “differential” collisional rate <p(e, i, b)> given by <p(e, i, b)>=(1/(2π)^2)∫p_col (e, i, b, τ, ω)dτdω. (24) Here, we calculated <p(e, i, b)> separately for 1-, 2-, and more recurrent orbits. The results are shown in Fig. 9, from which we can see that 2-recurrent collision orbits exist for relatively large b, and n-current (n≧3) ones exist only for b≒b_max. That is, the recurrent collision orbits appear only in cases of relatively low energy. From Eq. (10), we have <P(e, i)>=∫【-∞→∞】(3/2)|b|<p(e, i, b)>db. (25) Using evaluated values of <p(e, i, b)> for various b, we finally obtain <P(e, i)>=0.114 for (e, i)=(1.0, 0.5); the contribution of 2-recurrent orbits is 5%, and that of 3- and more-recurrent orbits is less than 1%. For this case (e=1.0 and i=0.5), we observed 874 collision orbits. The statistical error in evaluating <P(e, i)> is therefore presumed to be of the order of 4%. Since the contribution of 3- and more-recurrent orbits is within the statistical fluctuation, it can be neglected. よろしくお願いします。
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5. Normalization of collisional rate First, we introduce an enhancement factor defined as the ratio of the collisional rate <P(e, i)> to that in the two-body approximation <P(e, i)>_2B: R(e, i)= <P(e, i)>/ <P(e, i)>_2B (27) The factor R(e, i) gives a measure of the collisional rate enhancement due to the effect of solar gravity. In the two-dimensional case, <P(e,0)> is given by Eq. (11) while <P(e, 0)>_2B is defined by <P(e,0)>_2B=(2/π)E(√(3/4))ρ_(2D)v, (28) where E(k) is the second kind complete elliptic integral and ρ_(2D)v is given by Eq. (3) with <e(2/2)> replaced by e^2 (note that the units are changed, i.e., v=(e^2+i^2)^(1/2) and Gm_p=3). The numerical coefficient 2E(k)/π(=0.77) is introduced so that the collisional rate <P(e,0)>_2B coincides with <P(e,0)> in the high energy limit, v→∞ (see Paper I and Greenzweig and Lissauer, 1989). In the three-dimensional case, <P(e,i)> is given by Eq. (10) while <P(e, i)>_2B by Eq. (1) with <e(2/2) > and <i(2/2)> replaced, respectively, by e^2 and i^2. It should be noticed that <P(e,i)> has the dimension per unit surface number density n_s. Then, we define <P(e,i)>_2B by nσv/n_s; (n_s/n) corresponds to twice the scale height (in the z-direction) of a swarm of planetesimals. Usually, the scale height is taken to be i*a_0* (i.e., i, in the units here). As in the two-dimensional case, we require that <P(e,i)>_2B must coincide with <P(e,i)> in the high energy limit. Then, by introducing the numerical coefficient (2/π)^2E(k) (=0.49~0.64) (see Paper I), we have <P(e,i)>_2B=(2/π)^2E(k)πr_p^2{1+(6/(r_p(e^2+i^2)) }(e^2+i^2)^(1/2)/(2i), (29) with k^2=3e^2/4(e^2+i^2). (30) 6. The collisional rate for the two-dimensional case In this section, we concentrate on the collisional rate for the two-dimensional case where i=0. In this case, the small degrees of freedom of relative motion allow us to investigate in detail behaviors of orbital motion: it is sufficient to find collision orbits only in the b-τ two-dimensional phase space for each e, as seen in Eq. (11). 長文ですが、よろしくお願いします。
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The numbers of collision orbits found in the present calculations are shown in Table 4 for the representative sets of (e,i). From these numbers we can expect the magnitude of statistical error in the evaluation of <P(e,i)> to be a few percent for small e, i and within 10% for large e, i for r_p=0.005 are shown in Table 5, together with those of the two-dimensional case. Interpolating these values, we have obtained the contour of <P(e,i)> and R(e,i) on the e-I plane. They are shown in Figs. 14 and 15. From Fig. 15 we can read out the general properties of the collisional rate in the three-dimensional case: (i) <P(e,i)> is enhanced over <P(e,i)>_2B except for small e and i, (ii) <P(e,i)> reduces to <P(e,i)>_2B for (e^2+i^2)^(1/2)≧4, and (iii) there are two peaks in R(e,i) near regions where e≒1 (i<1) and where i≒3 (e<0.1): the peak value is at most as large as 5. In the vicinity of small v(=(e^2+i^2)^(1/2)) and i, R(e,i) rapidly reduces to zero. This is due to a singularity of <P(e,i)>_2B at v=0 and i=0 in the ordinary expression given by Eq. (29) and hence unphysical; the behavior of collisional rate in the vicinity of small v and i will be discussed in detail later. Thus, we are able to assert, more strongly, the property (i) mentioned in the last paragraph: that is, solar gravity always enhances the collisional rate over that of the two-body approximation. One of the remarkable features of R(e,i) found in Fig. 15 is the property (ii). That is, the collisional rate between Keplerian particles is well described by the two-body approximation, for (e^2+i^2)^(1/2)≧4. This is corresponding to the two-dimensional result that R(e,0)≒1 for e≧4. よろしくお願いします。
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Table 3. Numbers of two-dimensional collision orbits (i=0) found by our orbital calculations. Only four cases of e are tabulated as examples. The numbers of collision orbits decrease with the decrease in the planetary radius r_p. Fig.11. The two-dimensional enhancement factor R(e,0) as a function of e for r_p=0.005, 0.001, and 0.0002. The enhancement factor depends rather weakly on r_p. Fig.12. The collisional flux F(e,E) defined by Eq. (32) as a function of the Jacobi energy E for e=0, 0.5, 1.0, and 2.0. ↓Table 3. http://www.fastpic.jp/images.php?file=0416143752.jpg ↓Fig.11. http://www.fastpic.jp/images.php?file=9556242912.jpg ↓Fig.12. http://www.fastpic.jp/images.php?file=1594984078.jpg よろしくお願いします。
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Finally, we will add a comment on comparison of our result with those of Wetherill and Cox (1985). Wetherill and Cox examined three-dimensional calculation for a swarm of planetesimals with a special distribution, i.e., e_2 has one value and i_2 is distributed randomly between 0.3e_2 and 0.7e_2 (<i_2>=e_2/2) while e_1=i_1=0, which corresponds, in our notation of Eq. (9), to <n_2>={n_sδ(e_2-e)δ(i_2-i)/0.4π^2e(2/2) for 0.3e_2<i_2<0.7e_2, {0 otherwise. (38) Integrating <P(e,i)> with above <n_2> according to Eq. (9), we compare our results with theirs. Figure 18 shows that their results almost agree with ours (the slight quantitative difference may come from the difference in definition of the enhancement factor); but their results contain a large statistical uncertainty because they calculated only 10~35 collision orbits for each set of e and i while 100~6000 collision orbits were found in our calculation (see Table 4). Furthermore, our results are more general than theirs in the sense that their calculations are restricted to the special distribution of planetesimals as mentioned above, while the collisional rate for an arbitrary planetesimal distribution can be deduced from our results. 8. Concluding remarks Based on the efficient numerical procedures to find collision orbits developed in Sect. 2 to 4, we have evaluated numerically the collisional rate defined by Eq. (10). The results are summarized as follows: (i) the collisional rate <P(e,i)> is like that in the two-dimensional case for i≦0.1 (when e≦0.2) and i≦0.02/e (when e≧0.2), (ii) except for such two-dimensional region, <P(e,i)> is always enhanced by the solar gravity, (iii) <P(e,i)> reduces to <P(e,i)>_2B for (e^2+i^2)^(1/2)≧4, where <P(e,i)>_2B is the collisional rate in the two-body approximation, and (iv) there are two notable peaks in <P(e,i)>/<P(e,i)>_2B at e≒1 (i<1) and i≒3 (e<0.1); but the peak value is at most 4 to 5. From the present numerical evaluation of <P(e,i)>, we have also found an approximate formula for <P(e,i)>, which can reproduce <P(e,i)> within a factor 5 but cannot express the peaks found at e≒1 (i<1) and i≒3 (e<0.1). These peaks are characteristic to the three-body problem. They are very important for the study of planetary growth, since they are closely related to the runaway growth of the protoplanet, as discussed by Wetherill and Cox (1985). This will be considered in the next paper (Ohtsuki and Ida, 1989), based on the results obtained in the present paper. Acknowledgements. Numerical calculations were made by HITAC M-680 of the Computer Center of the University of Tokyo. This work was supported by the Grant-in-Aid for Scientific Research on Priority Area (Nos. 62611006 and 63611006) of the Ministry of Education, Science and Culture of Japan. Fig. 18. Comparison of the enhancement factors with those of Wetherill and Cox (1985). The error bars in their results arise from a small number (10~35) of collision orbits which they found for each e. Our results are averaged by the distribution function which they used (see text). Fig. 18.↓ http://www.fastpic.jp/images.php?file=0990654048.jpg かなりの長文になりますが、どうかよろしくお願いします。
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The second feature seen from Fig.11 is that the profile of R(e,0) does not depend significantly on r_p (for r_p=0.005 to 0.0002). Only an exception is found near e≒1, but this is, in some sense, a singular point in R(e,0), which appears in a narrow region around e≒1 ( in fact, for e=0.9 and 1.2, there is no appreciable difference between r_p=0.005 and 0.0002). Thus, neglecting such fine structures in R(e,0), we can conclude that R(e,0) does depend very weakly on r_p. In other words, the dependence on r_p of <P(e,0)> is well approximated by that of <P(e,0)>_2B given by Eq. (28). Now, we will phenomenalogically show what physical quantity is related to the peak at e≒1. We introduce the collisional flux F(e,E) for orbits with e and E, where E is the Jacobi energy given by (see Eq. (15)) E=e^2/2-(3b^2)/8+9/2. (31) The collisional flux F(e,E) is defined by F(e,E)=(2/π)∫【‐π→π】p_col(e,i=0, b(E), τ)dτ. (32) From Eqs. (11) and (31), we obtain <P(e,0)>=∫F(e,E)dE. (33) In Fig.12, F(e,E) is plotted as a function of E for the cases of e=0, 0.5, 1.0, and 2.0. We can see from this figure that in the case of e=1 a large fraction of low energy planetesimals contributes to the collisional rate compared to other cases (even to the cases with e<1). In general, in the case of high energy a solution for the three-body problem can be well described by the two-body approximation: in other words, in the case of low energy a large difference would exist between a solution for the three-body problem and that in the two-body approximation. As shown before, this difference appears as an enhancement of the collisional rate. Thereby an enhancement factor peak is formed at e≒1 where a large fraction of low-energy planetesimals contributes to the collisional rate. よろしくお願いいたします。
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3. Results of the orbital calculations In this paper, we describe our numerical results only for the special cases e_i~=0 and e_i~=4 in order to find the characteristic features of the two-body encounters of Keplerian particles. 3.1. For the case e_i~=0 In this case the orbital element, δ_i, loses its meaning because the particle orbits have no periheria and , hence, we can actually assign the initial condition by one parameter, b_i~. The orbital calculations are performed for about 3800 cases with various values of b_i~ from －10 to 10. よろしくお願いします。
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As mentioned above, there are two peaks in R(e,i) in the e-i diagram: one is at e≒1 (i<1) and the other at i≒3 (e<0.1). The former corresponds directly to the peak in R(e,0) at e≒1 found in the two-dimensional case. The latter is due to the peculiar nature to the three-dimensional case, as understood in the following way. Let us introduce r_min (i, b,ω) in the case of e=0, which is the minimum distance during encounter between the protoplanet and a planetesimal with orbital elements i, b, and ω. In Fig. 16, r_min(i,b)=min_ω{r_min(i,b,ω)} is plotted as a function of b for various i, where r_min<r_p (=0.005) means “collision”; there are two main collision bands at b≒2.1 and 2.4 for i=0. For i≦2, these bands still exist, shifting slightly to small b. This shift is because a planetesimal feels less gravitational attracting force of the protoplanet as i increases. As i increases further, the bands approach each other, and finally coalesce into one large collision band at i≒3.0; this large band vanish when i≧4. In this way, the peculiar orbital behavior of three-body problem makes the peak at i≒3 (e<0.1). Though there are the peaks in R(e,i), the peak values are not so large: at most it is as large as 5. This shows that the collisional rate is well described by that of the two-body approximation <P(e,i)>_2B except for in the vicinity of v, i→0 if we neglect a difference of a factor of 5. Now we propose a modified form of <P(e,i)>_2B which well approximates the calculated collisional rate even in the limit of v, i→0. We find in Fig.14 that <P(e,i)> is almost independent of i, i.e., it behaves two-dimensionally for i≦{0.1 (when e≦0.2), {0.02/e (when e≧0.2). (35) This transition from three-dimensional behavior to two-dimensional behavior comes from the fact that the isotropy of direction of incident particles breaks down for the case of very small i (the expression <P(e,i)>_2B given by Eq. (29) assumes the isotropy). In other words, as an order of magnitude, the scale height of planetesimals becomes smaller than the gravitational radius r_G=σ_2D/2 (σ_2D given by Eq. (3)) and the number density of planetesimals cannot be uniform within a slab with a thickness σ_2D for small i. Table 4. Numbers of three-dimensional collision events found by orbital calculations for the representative sets of e and i. In the table r_p is the radius of the protoplanet. Table 5. The three-dimensional collision rate <P(e,i)> for the case of r_p=0.005 (r_p being the protoplanetary radius), together with two-dimensional <P(e,i=0)> Fig. 14. Contours of the evaluated <P(e,i)>, drawn in terms of log_10<P(e,i)> Fig. 15. Contours of the enhancement factor R(e,i) Table 4.↓ http://www.fastpic.jp/images.php?file=1484661557.jpg Table 5.↓ http://www.fastpic.jp/images.php?file=6760884829.jpg Fig. 14. ＆Fig. 15.↓ http://www.fastpic.jp/images.php?file=8798441290.jpg 長文になりますが、よろしくお願いします。
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In the chaotic zone, there are, of course, a great number of discrete collision orbits. Minimum separation distance in the chaotic zone near b=1.93 is enlarged in Fig.6, which is obtained from the calculation of 3000 orbits with b between 1.926 and 1.932. Even in this enlarged figure, r_min varies violently with b. Although the chaotic zones are not sufficiently resolved in our present study, the phase space occupied by collision orbits in the chaotic zones is much smaller than that in the regular collision bands. Even if all orbits in the chaotic zone are collisional, their contribution to the collision rate is less than 4% of the total: the width in b=2.30 and 2.48, we also found that the total width is much smaller than 0.001. This implies that in the evaluation of <P(e, i)>, we can neglect the contribution of collision orbits in the chaotic zones. These are n-recurrent collision orbits in the regular zones. Of these, 2-recurrent collision orbits are most important. The collisional band composed of them is found near b=2.34. Its width ⊿b is about 0.011, and the contribution to the collision is as large as 15%. No.3- and more –recurrent collision orbits were observed in regular zones. They were found only in the chaotic zones and, hence, can be neglected. 長いですが、よろしくお願いします。
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